What does this failure look like in production?

In production this failure typically appears during the first tens-to-hundreds of training steps: loss curves become non-monotonic, validation metrics plateau above the baseline, and downstream checkpoints either diverge or fail to improve. Engineers frequently misread it as data drift when the underlying mechanism is the analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). if the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. pytorch's autograd engine will then propagate nan back to the network's weights.. Validating against the symptom list in this entry is the fastest way to confirm the diagnosis.

Is this a hardware or a software issue?

Mathematics failures can look like either layer depending on the symptom. The discriminator is the resolution of the symptom to the recommended fix: if applying the steps under "How to fix it" eliminates the symptom, this is a software/configuration class issue regardless of the hardware it surfaced on. Repeat the fix on a fresh node to confirm.

How long does the fix typically take?

Most resolutions from this entry take 10 to 30 minutes of engineer time once the diagnosis is confirmed. Long fixes come from skipping the diagnosis step and chasing apparent symptoms (validation drift, memory growth, slow throughput) instead of the actual root cause listed above. Apply the fix in a small experiment first, then roll it forward.

Does this affect inference or only training?

Primarily this is a mathematics failure observed during training, but the same misconfiguration can leak into downstream evaluation and serving. Treat the prevention checklist as a guardrail: any pipeline that touches a trained checkpoint should re-validate the failure mode before promoting the model.

What's the cheapest way to confirm this is the problem?

Cheapest signal: reproduce the symptom on a 1-GPU toy run with the same configuration. If the toy run exhibits the same behavior, the cause is the configuration rather than the cluster. If only the multi-GPU run is affected, the failure is a distributed interaction, so PyTorch is the most likely layer to inspect.

Can the Denpex platform detect this automatically?

Yes. Denpex correlates the symptoms in this entry against your training logs in real time and surfaces the failure taxonomy, root cause, and exact remediation command on the console. Three full diagnoses are included free during signup, and the encyclopedia entry you are reading is generated from the same signal library that powers the platform.

What related failures should I check next?

Closely related failures cluster around #Function Singularity. After applying this fix, validate against the related entries below to confirm no overlapping cause is masking as this one. Engineers routinely discover that the dominant failure mode is actually a second, sibling failure compounded on top.

Will this recur after the fix?

Applied once with the prevention checklist, this failure does not recur under normal operation. The recurring variant most engineers encounter is the re-emergence of the same symptom on a different model or stack after Function Singularity is upgraded. Pin compatible versions and rerun the prevention smoke test on every dependency change.

Is there a fingerprint I can monitor?

Yes. Forward pass outputs valid numbers, but loss or gradients become NaN on the backward pass. is the highest-signal fingerprint. Wire a metric or alert on that signal (or its absence after the fix) so you can detect the recurrence automatically. The Denpex platform encodes this exact fingerprint and surfaces it directly in the dashboard.

What is the official documentation that covers this?

Reference documentation is listed in the References section at the bottom of this entry. The most authoritative URLs are the upstream framework docs and the issue trackers cited; the Denpex entry synthesizes both into the executive, engineer, and on-call perspectives so each audience can act without reading every linked source.

Does this entry include working code?

Yes. The "Code Examples" section contains copy-paste-ready snippets that target the most common stacks (PyTorch, Hugging Face Trainer, DeepSpeed, Megatron-LM, vLLM/Transformers). Each snippet is annotated with why it works so you can adapt the pattern to your own framework without losing fidelity.

Autograd Singularity at sqrt(0)

Q: How does this scale from 1 GPU to 1000 GPUs?

At single-GPU scale the symptom usually looks like slow convergence or a single anomalous loss value. At 64+ GPU scale, distributed interactions amplify the analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). if the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. pytorch's autograd engine will then propagate nan back to the network's weights. through collectives, and the failure can manifest as a full job abort. Always reproduce the fix at the smallest cluster configuration that still exhibits the symptom before promoting to production scale.

The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights.

Quick answer

The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)).

Symptom: RuntimeError: Function 'SqrtBackward' returned nan values
Root cause: The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights.
Recommended fix: Add a small epsilon before the square root out = torch.sqrt(x + 1e-8) Adding a small constant ensures the input to the derivative function is never strictly zero, avoiding the singularity.
How Denpex helps: Denpex matches Autograd Singularity at sqrt(0) across every rank in a distributed run and reports which rank failed first, so you act on the initiating node instead of the loudest one.

Mathematics#Function Singularity

What this failure is

Autograd Singularity at sqrt(0) is a Mathematics failure seen during ML training runs. The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights. Common tags: Function Singularity.

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Why it happens (the mechanism)

The user sees no warnings during the forward pass because `torch.sqrt(0.0)` is completely valid and equals `0.0`. It only crashes during `backward()`, making it seem like a gradient accumulation bug.

What you'll observe

RuntimeError: Function 'SqrtBackward' returned nan values

Common symptoms and what they mean

Symptom	Why it happens
Forward pass outputs valid numbers, but loss or gradients become NaN on the backward pass.	The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights.
Issue occurs randomly, often when calculating pairwise distances or L2 norms that happen to perfectly align (distance = 0).	The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights.

Which systems are affected

PyTorch
CUDA

How to confirm this is the problem

Use this checklist to test the hypothesis against a small reproduction. No single line proves the root cause, so preserve the preceding events and compare one variable at a time.

✓Run script with `torch.autograd.set_detect_anomaly(True)` to pinpoint the exact sqrt operation.
✓Check input tensors to `torch.sqrt()` for zero values.

Searchable error signature

search key

RuntimeError: Function 'SqrtBackward' returned nan values

Use this text as a lookup key in logs and upstream issue trackers. It is not presented as a captured customer log. Confirm the cause from your own preceding events, versions, configuration and the cited references.

The fix and the prevention pattern

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Root cause

The analytical derivative of sqrt(x) is 1 / (2 * sqrt(x)). If the input x evaluates to exactly 0, the denominator becomes 0, and the gradient evaluates to infinity. PyTorch's autograd engine will then propagate NaN back to the network's weights.

The fix and how to prevent it

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References

↗PyTorch Forums: NaN gradients from sqrt

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